论文标题
在三个球定理上用于Helmholtz方程的解决方案
On the three ball theorem for solutions of the Helmholtz equation
论文作者
论文摘要
令$ u_k $成为helmholtz方程的解决方案，带有波数$ k $，$Δu_k+k^2 u_k = 0 $，在$ \ m \ m}^n $，$ \ m}^n $中的小球上对于固定点$ p $，我们定义$ m_ {u_k}（r）= \ max_ {d（x，x，p）\ le r} | u_k（x）|。$以下三个球不等式$ m_ {u_k}（u_k}（u_k}（2r）\ le c（k，r，α）m_ {u_k}（r）^αm_{u_k}（4r）^{1-α} $是众所周知的，它可容纳一些$α\ in（0,1）$和$ c（k，r，r，α）> 0 $ u_k $。我们表明，常数$ c（k，r，α）$在$ k $中成倍增长（当$ r $固定而小时）。我们还将我们的结果与Riemannian歧管上Helmholtz方程的Cauchy问题解决方案的稳定性进行了比较。
Let $u_k$ be a solution of the Helmholtz equation with the wave number $k$, $Δu_k+k^2 u_k=0$, on a small ball in either $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{H}^n$. For a fixed point $p$, we define $M_{u_k}(r)=\max_{d(x,p)\le r}|u_k(x)|.$ The following three ball inequality $M_{u_k}(2r)\le C(k,r,α)M_{u_k}(r)^αM_{u_k}(4r)^{1-α}$ is well known, it holds for some $α\in (0,1)$ and $C(k,r,α)>0$ independent of $u_k$. We show that the constant $C(k,r,α)$ grows exponentially in $k$ (when $r$ is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds.
